# Why we say that a probability measure $P$ is defined on $(\Omega, \mathcal{F})$?

I was wondering why we say that a probability measure $$P$$ is defined on $$(\Omega, \mathcal{F})$$, being $$\Omega$$ the sample space and $$\mathcal{F}$$ one sigma-algebra, when actually we have that $$P$$ is a mapping $$P:\mathcal{F}\rightarrow [0,1]$$. I am not very familiar with the measurement theory and maybe that's why I am asking this (it could be a very trivial question)... But given that actually $$\Omega\in\mathcal{F}$$ why not to say that $$P$$ is defined on $$\mathcal{F}$$, I don't see the point of including $$\Omega$$ explicitly. Thanks in advance, and forgive my ignorance on the matter.

• You are correct in that $P$ is defined on the measurable subsets of $\Omega$, that is, the elements of $\mathcal F$, so $(\Omega,\mathcal F)$ is slightly redundant, but this is the traditional wording to insist upon the dependence on the $\sigma$-algebra. – Xi'an Mar 27 at 9:02